The image depicts an angle system with intersecting lines at point $Q$, where several angle expressions are given. Let's analyze and solve it:

Angles Given:

1. $\angle PQS = (5x + 16)^\circ$
2. $\angle PQR = (3x + 12)^\circ$
3. $\angle SQT = (6y - 21)^\circ$

Since the angles $\angle PQS$ and $\angle PQR$ are on a straight line (a linear pair), the sum of these two angles must equal $180^\circ$ because they form a straight angle.

Equation 1 (for $x$):

$(5x + 16) + (3x + 12) = 180$ Simplifying: $8x + 28 = 180$ $8x = 152$ $x = 19$

Next, solving for $y$:

The angle $\angle SQT$ and $\angle PQS$ appear to be vertical angles, which are always congruent (equal).

Thus, we have: $5x + 16 = 6y - 21$ Substituting $x = 19$ from the previous solution: $5(19) + 16 = 6y - 21$ $95 + 16 = 6y - 21$ $111 = 6y - 21$ $6y = 132$ $y = 22$

Final values:

• $x = 19$
• $y = 22$

Would you like further explanation or details on this solution?

Here are 5 related questions:

1. How are linear pairs of angles used to set up equations in geometry?
2. What are vertical angles, and why are they always equal?
3. Can you solve a similar problem involving different angle relationships?
4. How would the solution change if one angle were given instead of an expression?
5. What real-life applications involve solving angles?

Tip: Always look for linear pairs and vertical angles when dealing with intersecting lines, as these relationships simplify many problems.